Are groups of different orders necessarily non-isomorphic?

Question

I am trying to teach myself group theory and I recently came across the topic of Isomorphisms. I know that 2 groups are isomorphic if there is a one-on-one correspondence between their elements. So if the groups have a different order, does that mean they are not isomorphic? Such as a group $S$ and its permutation group $S_{n}$ for $n>1$.

Answer 1

The answer is “yes”, but your definition of isomorphism is not correct. An isomorphism between groups is a bijection $\varphi$ which preserves the product, that is, such that $\varphi(x.y)=\varphi(x).\varphi(y)$ for every $x$ and every $y$ in the domain. But since it must be a bijection then, yes, groups with distinct orders cannot be isomorphic.